Flow over a non-uniform sheet with non-uniform stretching (shrinking) and porous velocities

Viscous flow over a porous and stretching (shrinking) surface of an arbitrary shape is investigated in this article. New dimensions of the modeled problem are explored through the existing mathematical analogies in such a way that it generalizes the classical simulations. The latest principles provide a framework for unification, and the consolidated approach modifies the classical formulations. A realistic model is presented with new features in order to explain variety of previous observations on the said problems. As a result, new and upgraded version of the problem is appeared for all such models. A set of new, unusual, and generalized transformations is formed for the velocity components and similarity variables. The modified transformations are equipped with generalized stretching (shrinking), porous velocities, and surface geometry. The boundary layer governing equations are reduced into a set of ordinary differential equations (ODEs) by using the unification procedure and technique. The set of ODEs has two unknown functions f and g. The modeled equations have five different parameters, which help us to reduce the problem into all previous formulations. The problem is solved analytically and numerically. The current simulation and its solutions are also compared with existing models for specific value of the parameters, and excellent agreement is found between the solutions.